Lecture 16 Maximum Matching. Incremental Method Transform from a feasible solution to another feasible solution to increase (or decrease) the value of.

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Presentation transcript:

Lecture 16 Maximum Matching

Incremental Method Transform from a feasible solution to another feasible solution to increase (or decrease) the value of objective function.

Matching in Bipartite Graph Maximum Matching

1 1

Note: Every edge has capacity 1.

2. Can we do those augmentation in the same time? 1. Can we do augmentation directly in bipartite graph?

1. Can we do augmentation directly in bipartite graph? Yes!!!

Alternative Path

Optimality Condition

Puzzle

Extension to Graph

Matching in Graph Maximum Matching

Note We cannot transform Maximum Matching in Graph into a maximum flow problem. However, we can solve it with augmenting path method.

Alternative Path

Optimality Condition

2. Can we do those augmentation in the same time?

Hopcroft–Karp algorithm The Hopcroft–Karp algorithm may therefore be seen as an adaptation of the Edmonds-Karp algorithm for maximum flow. Edmonds-Karp algorithm

In Each Phase

Running Time Reading Material

Max Weighted Matching

Maximum Weight Matching It is hard to be transformed to maximum flow!!!

Minimum Weight Matching

Augmenting Path

Optimality Condition

35

36

37

38

39

40

41

42

43

44

Chinese Postman

Minimum Weight Perfect Matching Minimum Weight Perfect Matching can be transformed to Maximum Weight Matching. Chinese Postman Problem is equivalent to Minimum Weight Perfect Matching in graph on odd nodes.